REAL‐TIMEWEAKNESS OFTHEGLOBALECONOMY · Contents Abstract 4 1.Introduction 5...

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DANILO LEIVA‐LEON | GABRIEL PEREZ‐QUIROS | EYNO ROTS REAL‐TIME WEAKNESS OF THE GLOBAL ECONOMY MNB WORKING PAPERS | 4 2020 JULY

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Page 1: REAL‐TIMEWEAKNESS OFTHEGLOBALECONOMY · Contents Abstract 4 1.Introduction 5 2.InferringHeterogeneousRecessions 7 3.AssessingWeaknessAcrossCountries 10 3.1. Data 10 3.2. TheCaseoftheUnitedStates

DANILO LEIVA‐LEON | GABRIEL PEREZ‐QUIROS | EYNO ROTS

REAL‐TIME WEAKNESS

OF THE GLOBAL ECONOMY

MNB WORKING PAPERS | 4

2020J U L Y

MNB WORKING PAPERS 4 • 2020 I

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MAGYAR NEMZETI BANK

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REAL‐TIME WEAKNESS

OF THE GLOBAL ECONOMY

MNB WORKING PAPERS | 4

2020J U L Y

MNB WORKING PAPERS 4 • 2020 III

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MAGYAR NEMZETI BANK

The views expressed are those of the authors’ and do not necessarily reflect the official view of the central bank of Hungary

(Magyar Nemzeti Bank).

MNB Working Papers 2020/4

Real‐Time Weakness of the Global Economy*

(Valós Idejű Globális Sérülékenységi Index)

Written by Danilo Leiva‐Leon, Gabriel Perez‐Quiros, Eyno Rots

Budapest, July 2020

Published by the Magyar Nemzeti Bank

Publisher in charge: Eszter Hergár

Szabadság tér 9., H‐1054 Budapest

www.mnb.hu

ISSN 1585‐5600 (online)

*Danilo Leiva‐Leon, Banco de España, [email protected]; Gabriel Perez‐Quiros, European Central Bank and CEPR,[email protected]; Eyno Rots, Magyar Nemzeti Bank, [email protected]. We would like to thank our col‐leagues of the ESCB Expert Group onNonlinearModels for their stimulating and helpful comments, and an anonymous refereefor suggestions. We also thank Romain Aumond for excellent research assistance. The views expressed in this paper are thoseof the authors and are in no way the responsibility of the Banco de España, European Central Bank, Eurosystem, or MagyarNemzeti Bank.

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Contents

Abstract 4

1. Introduction 5

2. Inferring Heterogeneous Recessions 7

3. Assessing Weakness Across Countries 10

3.1. Data 10

3.2. The Case of the United States 10

3.3. The Case of the Euro Area¹ 14

3.4. Other Advanced Economies 14

3.5. Emerging Markets 15

4. Assessing Global Weakness 17

4.1. Dynamics and Sources of Weakness 17

4.2. Monitoring Global Risks 21

5. Conclusions 26

Appendix A. Model and Estimation 29

A.1. Full specification of the model 29

A.2. Bayesian estimation 31

Appendix B. Supplementary Figures 35

¹ The Euro Area model has been estimated in the context of the task force of the Eurosystem on nonlinear tools.

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Abstract

We propose an empirical framework to measure the degree of weakness of the global economy in real‐time. It relies on non‐linear factor models designed to infer recessionary episodes of heterogeneous deepness, and fitted to the largest advancedeconomies (U.S., Euro Area, Japan, U.K., Canada and Australia) and emerging markets (China, India, Russia, Brazil, Mexico andSouth Africa). Based on such inferences, we construct a Global Weakness Index that has three main features. First, it can beupdated as soon as new regional data is released, as we show by measuring the economic effects of coronavirus. Second, itprovides a consistent narrative of the main regional contributors of world economy’s weakness. Third, it allows to performrobust risk assessments based on the probability that the level of global weakness would exceed a certain threshold of interestin every period of time. With information up to March 2nd 2020, we show that the Global Weakness Index already sharplyincreased at a speed at least comparable to the experienced in the 2008 crisis.

JEL: E32, C22, E27.

Keywords: International, Business Cycles, Factor Model, Nonlinear.

Összefoglaló

A globális gazdaság sérülékenységét vizsgáljuk egy empírikus modellel valós idejű adatokon. Fejlődő (Kína, India, Oroszország,Brazília, Mexikó, és Dél‐afrikai Köztársaság) és fejlett (USA, eurózóna, Japán, Egyesült Királyság, Kanada, és Ausztrália) orszá‐gok adataira illesztett nem‐lineáris faktor modellel vizsgáljuk a recessziók idején adott heterogén válaszokat. A modellből nyertinformációkból összeállítunk egy Globális Sérülékenységi Indexet aminek három fő tulajdonsága van: i) friss régiós adatok beér‐kezésekor az index könnyen frissíthető és ezáltal a koronavírus gazdasági hatásai valós időben elemezhetővé válnak, ii) a globálisgazdasági kockázatok régiós mozgatórugóit egy konzisztens elemzési keretben vizsgálhatjuk, iii) a globális kockázatok emelke‐dése esetén a modellből számított globális kockázatokat jelző index meghalad egy előre meghatározott valószínűséget. 2020Március 2‐ig rendelkezésre álló adatokon megmutatjuk, hogy a Globális Sérülékenységi Index a 2008‐as pénzügyi válság idejéntapasztalt sebességgel emelkedett.

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1 Introduction

The last Global Financial Crisis has had ramifications that, arguably, can be seen to this day across the globe. Europe has been noexception: it has witnessed a dramatic recession in 2008–2009 across the euro area and the Sovereign Debt Crisis in 2009–2012;ever since, it has been plagued by lacklustre growth that has turned resistant to a plethora of unprecedented non‐conventionalmonetary and fiscal stimuli. More recently, there has been growing concern among academics and policy‐makers about a newrecessionary phase. This concern is not confined to U.S. and the Euro Area. Based on the latest adverse global economicdevelopments, influenced by the outbreak of a new disease in China associated to a new coronavirus, and spread around theglobe, international organizations, such as the IMF and the OECD, have downgraded to outlook of the world economy.² Giventhe looming danger of a new global downturn, the need for a framework to infer the strength of the world economy in real‐timeis as high as ever.

Given two of themain defining characteristics of business cycles, which are comovement across real activity indicators and non‐linear dynamics resembling up and downturns (Burns and Mitchell, 1946), previous works have proposed econometric frame‐works that account for these features when inferring recessionary episodes. In particular, Markov‐switching dynamic factor(MSDF) models have been successfully used to account for comovements and nonlinearities in a unified setting. Introduced byChauvet (1998), MSDF models where initially applied to a set of U.S. real activity indicators at the monthly frequency with theaim of summarizing such information into a single index subject to regime changes, showing its ability to identify turning pointsin a timely fashion.³ Other works have focused on extending such a framework to operate in the context of mixed‐frequencydata, to include information on quarterly real GDP (Camacho et al., 2014) or on nominal GDP (Barnett et al., 2016).

In the context ofMSDFmodels, the common factor summarizing the information in a set of activity indicators is assumed to haveanunconditionalmean associated to expansions, 𝜇exp, and another unconditionalmean associated to recessions, 𝜇rec. FollowingHamilton (1989), previous MSDF models have assumed that 𝜇exp holds for all the expansions, and that 𝜇rec also holds for allrecessionary episodes, included in the sample. However, this assumption can be highly restrictive, in particular, if recessions areconsiderably heterogeneous over time in terms of deepness. For example, assuming that 𝜇rec does not vary across recessionscould preclude the model to accurately infer an upcoming ‘mild’ recession after having only observed a ‘severe’ recession, ashappened in most advanced economies after the Global Financial Crisis. Hence, although MSDF models allow for a timelyassessment of turning points by relying on a set of indicators, they might be subject to a lack of accuracy when implementedin a context of heterogeneous downturns, which is typically observed at the international level. Jerzmanowski (2006) showsthat output growth of emerging economies exhibit substantially different types of recurrent recessionary regimes. Also, Aguiarand Gopinath (2007) illustrate that modelling business cycles nonlinearities associated to emerging markets tends to be evenmore challenging than for the case of developed economies.

In a more recent work, Eo and Kim (2016) propose a model that uses real GDP growth data to produce inferences of U.S. reces‐sions by taking into account their heterogeneity over time. The model consists of a univariate Markov‐switching specificationsubject to time‐varying means (MSTM). In particular, the authors assume that quarterly GDP growth has a mean 𝜇exp,𝜏0 , asso‐ciated to the 𝜏0‐th expansion, and a mean 𝜇rec,𝜏1 , associated to the 𝜏1‐th recession, in the sample under consideration. Both ofthesemeans are random‐walk processes. That is, each expansionary and recessionary episode has its own correspondingmean,which in part depends on the past and in part comes from random shocks. While this feature certainly helps to accurately detectturning points in a context of heterogeneous downturns, this model only produces inferences for one low frequency variable,quarterly GDP growth. Such a feature precludes the MSTMmodel delivering a robust assessment on the state of the economy,based on a set of activity indicators, that can be updated in real‐time, which is of high relevance due to the rapidly changingeconomic environment.

² In February 22nd 2020 the IMF decreased 0.1% global growth, IMF (2020). Only a week later, on March 2nd the OECD decreased global growth in0.5%, OECD (2020).

³ Chauvet and Piger (2008) rely on a similar model, however, it is estimated with Bayesian instead of classical methods.

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MAGYAR NEMZETI BANK

The aim of this paper is developing a flexible empirical framework, that accounts for downturns of heterogeneous deepnessand that also can be used in a real‐time environment, to provide robust assessments on the strength of the global economy.Such assessments are based on information gathered from the largest world’s economies, both advanced and emerging. Indoing so, we proceed in two steps. First, we design a nonlinear factor model that allows for mixed frequency data and time‐varying recession means, that is, we combine the multivariate setting associated to the MSDF model with the nonlinearitiesembedded in the MSTM model. By using data on economic indicators at the monthly and quarterly frequency, the modelis independently fitted to six advanced economies and six emerging markets. The selection of the countries is based on thesize of their economies, covering altogether more than seventy percent of the world GDP.⁴ Due to its flexibility in dealing withnonlinearities, our model is able to reproduce, for all the regions, timely and accurate inferences on regimes of weak activity,which are aligned with corresponding downturns in output growth. Second, the inferences associated to the twelve regionsare summarized into a single Global Weakness Index (henceforth GWI) that assesses the state of the world economy, and thatcan be updated on a daily basis, whenever new information on the regions is released. To the best of our knowledge, this isthe first model‐based index that uses economic data to provide updates on the state of the global economy with such a highfrequency.⁵

An important feature of the GWI is that it is bounded between zero and one, where values close to one represent highweaknessand values close to zero indicate lowweakness. This feature facilitates its interpretation and also comparisons between differentepisodes of interest. We show that the proposed index timely tracks periods where the global economy has been substantiallyweak, such as the period of the Great Recession, between the late 2015 and early 2016, and the present time. Also, we comparethe ability of our framework with non‐model‐based measures of the state of the world economy, finding that the GWI largelyleads the perception of agents about an upcoming global recession, proxied by information based on web searches.

Since the weakness at the global level is based on a weighted average of the weakness at the regional level, the GWI can bestraightforwardly decomposed into the time‐varying contributions associated to each region. Based on this decomposition, weprovide a narrative of the evolving strength of the global economy and its main contributors. In particular, we quantify thesubstantial importance of U.S., by the end of 2007, in the deterioration of global economic conditions, and the large influencethat had emerging markets on the global recovery phase, in the late 2009. Lastly, the GWI is employed to monitor global risksin real‐time. This is carried out by computing the probability that the level of global weakness would exceed a certain thresholdof interest. This information is collected in a real‐time environment, providing the entire spectrum of risks associated to adownturn of the world economy.

Finally, when the paper was almost written, unexpected and adverse global economic development took place. In January2020 an outbreak of a new disease associated with the coronavirus (COVID‐19) hit the international stock markets and tradeactivity. Motivated by these events, we test the ability of the proposed methodology to detect global weaknesses in real timeby assesing the effect of the coronavirus outbreak on the global economy.

The paper is organized as follows. Section 2 proposes the empirical framework to infer recessions of heterogeneous deepness.Section 3 provides inferences about the weakness of the largest world economies under consideration. Section 4 introducesthe global weakness index and illustrates its main features. Section 5 concludes.

⁴ For advanced economies, we include U.S., Euro Area, Japan, U.K., Canada and Australia, and for emerging markets, China, India, Russia, Brazil, Mexicoand South Africa.

⁵ On a related work, in a linear framework, Kilian (2019) relies on information on shipping costs to proxy world economic activity. Although, Hamil‐ton (2019) concludes that data on world industrial production provides a better indication of global activity.

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2 Inferring HeterogeneousRecessions

In this section, we introduce a new class of dynamic factor models, in which the common factor follows Markov‐switchingdynamics that are subject to time‐varying means. The proposed model summarizes the information contained in a set of realactivity indicators into a common factor that accounts for heterogeneous recessions, and that can be interpreted as an indexthat proxies the business cycle dynamics of a given economy. For convenience of exposition, we first focus on describing thedynamics of the latent common factor, and then, we proceed to detail how such a latent factor is extracted from the observeddata.

We assume that the common factor, ft, follows nonlinear dynamics that are flexible enough to accommodate the realization ofrecessions of different magnitudes,

ft = 𝜇0(1− st) + 𝜇1st + stxt + ef,t, ef,t ∼ 𝒩(0, 𝜎2f ) i.i.d., (1)

where st ∈ {0, 1} is a latent discrete variable that equals 0 when the economy is in a ‘normal’ episode, and takes the value of1 when the economy faces an ‘abnormal’ episode. The variable st is assumed to follow a two‐state Markov chain defined bytransition probabilities:

Pr(st = j|st−1 = i, st−2 = h, ...) = Pr(st = j|st−1 = i) = pij. (2)

Notice that since there are two states, these probabilities can be summarized by the chance of remaining in a normal state, p,and the chance of remaining in an abnormal state, q.

The variable xt in Equation (1) is defined as another unobserved process that evolves over time as follows:

xt = stxt−1 + (1− st)vt, vt ∼ 𝒩(0, 𝜎2v ) i.i.d. (3)

This law of motion implies that during normal times, when st = 0, xt is a white noise which has no impact on the commonfactor ft. However, during an abnormal episode, when st = 1, the value of xt remains fixed and is passed to the common factor.Our specification essentially implies that the common factor is a white‐noise process: during normal times, ft ∼ 𝒩(𝜇0, 𝜎2

f ),and during a recession that has started in period 𝜏, ft ∼ 𝒩(𝜇1 + x𝜏−1, 𝜎2

f ).⁶ That is, the common factor has the same constantmean 𝜇0 in normal times, but each abnormal episode is unique in the sense that the common factor during such episode wouldhave a mean 𝜇1 adjusted by the value of x𝜏−1, which, in turn, is unique for each episode. Obviously, this value x𝜏−1 is estimatedfrom the observed data as the magnitude that better fits each recessionary period.

The novelty of the proposed nonlinear factor model is in the dynamics of the common factor, which in our case is a functionof a random variable that fluctuates around a state‐dependent mean to account for heterogeneous recessions. The mean ofthe common factor is at the same constant level in each period when the economy is in the normal state. However, when theeconomy switches to an abnormal state, the priormean is drawn froma randomdistribution and remains the same for the entireduration of the abnormal episode, until the economy reverts back to a normal episode. In other words, all normal episodescome with the same mean of the common factor, whereas each abnormal episode comes with its own unique common‐factormean.

The proposed specification falls between that of Hamilton (1989), in which the common‐factor means associated to normaland abnormal episodes are two constant values, and that of Eo and Kim (2016), in which the two means are separate random‐walk processes, which gradually evolve over time, allowing for different observed growth rates in times of both normal and

⁶We assume that the common factor is a white‐noise process despite the fact that some of the data shows persistence. Although we do capture someof the persistence by assuming AR dynamics for the idiosyncratic components (see equation 7), it is definitely a possibility to consider the case whensome of the persistent behavior is shared by all the observed variables and assume a persistent common factor.

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MAGYAR NEMZETI BANK

abnormal episodes. We justify our modelling choice as the minimal specification that is necessary to account for the obviousfact that recessionary episodes come with very different growth rates of the GDP, and other real activity indicators. On theone hand, the Global Financial Crisis of 2007–2008 was unusually severe for many economies that, compared to it, subsequentrecessions look barely distinguishable from normal times. For example, based on the data for many European countries, onewould almost certainly fail to detect any other recession but the one observed upon the Global Financial Crisis when using aMarkov‐switchingmodel with the common factor of the Hamilton type. On the other hand, formany countries, there are so fewrecessionary episodes observed in the available data that a rich model that allows for independent recession‐ and expansion‐specific common factor means of the Eo‐Kim type would often be challenging to estimate without large model uncertainty. Themodel that we propose has a minimalistic structure due to potential lack of available data for the economies of interest, yet itis rich enough to account for the fact that every recession comes with a unique magnitude.

It is also important to note that an estimated Markov‐switching model has its own internal definition of states, which arealmost surely not aligned with the existing formal definitions of recessions and expansions. One particular concern is that fora decline in real activity to be called a recession, it is a typical requirement that the downturn lasts for some time, such as twoquarters. One approach to make the low‐growth state of the estimated model better correspond to a recession is to impose arequirement that a low‐growth episode must be persistent. For example, we could impose a restriction on minimum regimeduration as in Billio et al. (2016), or introduce duration dependence of the switching probabilities as in Kim and Nelson (1998).Rather, we estimate the model without such restrictions and allow it to have its own internal interpretation of high and lowgrowth regimes. Then, it is possible to use the posterior simulations to simulate the GDP growth for two quarters ahead andestimate the probability that the GDP growth will be negative for two quarters, if this is the definition of a recession. This way,the model is flexible about the regimes and at the same time capable of matching the required formal definitions of booms andrecessions.

Regarding the extraction of the factor from a set of observed information, each real activity indicator is assumed to be contem‐poraneously influenced by a common component and an idiosyncratic component. However, the treatment that each indicatorreceives depends on its corresponding frequency. In particular, indicators at the monthly frequency, ymi,t, can be expressed as:

ymi,t = 𝛾ift + ui,t, (4)

where 𝛾i denotes the associated factor loading and ui,t represents the idiosyncratic component. Instead, when dealing withindicators at the quarterly frequency, yqj,t, we follow Mariano and Murasawa (2003) and express quarter‐on‐quarter growthrates into month‐on‐month unobserved growth rates:

yqj,t =13yj,t +

23yj,t−1 + yj,t−2 +

23yj,t−3 +

13yj,t−4. (5)

Then, a quarterly growth rate can be expressed in terms of its idiosyncratic component and the common factor, as follows:

yqj,t = 𝛾j13ft +

23ft−1 + ft−2 +

23ft−3 +

13ft−4 + 1

3uj,t +

23uj,t−1 + uj,t−2 +

23uj,t−3 +

13uj,t−4. (6)

Lastly, the idiosyncratic components, ui,t, contain information that is exclusively associated to a particular indicator, after ac‐counting for its degree of commonality with the rest of variables. They are assumed to follow autoregressive dynamics oforder P,

ui,t = 𝜓i,1ui,t−1 + ... + 𝜓i,Pui,t−P + ei,t, ei,t ∼ 𝒩(0, 𝜎2i ) i.i.d. (7)

Given the nonlinearities embedded in themodel (1)‐(7), we rely on Bayesianmethods to produce inferences on both its param‐eters and latent variables. Let yt denote the vector of observed monthly and quarterly indicators, and let Y = {yt}Tt=1 contain allthe available data up to time T. Similarly, we define Z = {zt}Tt , where zt denotes a vector containing latent states correspondingto the common factor ft and idiosyncratic components ui,t. Also, let S = {st}Tt=1 be the collection of the latent regimes, and letX = {xt}Tt=1 contain the information on the unobserved adjustments to the mean growth rate during the abnormal episodes.All the parameters that specify the model are collected in:

𝜃 = p, q, 𝜇0, 𝜇1, 𝜎2f , 𝜎2

v , {𝛾i}, {𝜓i,m}, {𝜎2i } .

The model is estimated using a modification of the Carter and Kohn (1994) algorithm, which simulates, in turn, the unobservedstates Z, the unobserved mean adjustments X, and the indicators for abnormal episodes S. For a single run, given data Y andinitial guesses 𝜃0, X0, and S0, the following iterative procedure is used:

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INFERRING HETEROGENEOUS RECESSIONS

1. Given Y, Si−1, Xi−1 and 𝜃i−1, generate Zi from P(Z|Y, S, X, 𝜃). This step follows Appendix 1 of Carter and Kohn (1994) anduses the definition of the common factor (1) and the observation equations (4) and (6).

2. Given Zi, Xi−1, and 𝜃i−1, generate Si from P(S|Z, X, 𝜃). This step is based on equation (1) for the common factor and followsAppendix 2 of Carter and Kohn (1994).

3. Given Zi, Si−1, and 𝜃i−1, generate Xi from P(X|Z, S, 𝜃). This step follows the same procedure as in Step 1, but now equationfor the common factor (1) is treated as the observation equation, and equation (3) is treated as the law of motion for theunobserved state xt.

4. Given Y, Zi, Xi, Si, and standard conjugate priors, simulate 𝜃i from the standard posterior distributions.

A more detailed description of the model and of the simulation procedure is provided in the Appendix A.

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3 Assessing Weakness AcrossCountries

In this section, we evaluate the performance of the proposed econometric framework to infer low economic activity regimes,or ‘abnormal’ episodes, both from real‐time and international perspectives. First, we illustrate the advantages of allowing forrecession‐specific means in nonlinear factor models (referred to as time‐varying mean) by comparing their ability to anticipateturning points in real‐time with the one associated to a regular nonlinear factor model, which assumes the same mean acrossall recessionary episodes (referred to as constant mean). Next, we show that our framework is flexible enough to be used foreither advanced or emerging economies. The literature involving the use of this type of frameworks to infer turning pointsin emerging economies is scarce. One possible reason for this is that modelling business cycles nonlinearities associated toemerging markets tends to be more challenging than for the case of developed countries due to a variety of features, such asstrongly counter‐cyclical current accounts or dramatic ‘sudden stops’ in capital inflows (Aguiar and Gopinath, 2007), the role ofcountry risk in the determination of emerging countries interest rates (Neumeyer and Perri, 2005), or the role of patterns ofproduction and international trade (Kohn et al., 2018).

3.1 DATAThe proposed factor model, in equations (1)‐(7), is independently fitted to twelve of the largest world economic regions, whichtogether account for more than seventy percent of the world GDP. This set of regions include U.S., Euro Area, Japan, U.K.,Canada and Australia, among advanced economies and, China, India, Russia, Brazil, Mexico and South Africa, among emergingmarkets. The list of the variables employed for each region is reported in Table 1. In general, we follow the original approach ofStock andWatson (1991), Chauvet and Piger (2008) or Camacho et al. (2018), mimicking national accounts procedures. We usesupply side variables (usually, industrial production), internal demand variables (imports or sales), external demand (exports)and one additional variable, intrinsic of each economy. To all these monthly variables we add GDP. Those papers show thatthese small set of variables are reliable to address current conditions of the economy and comprise most relevant informationto infer recessions in real time.

In addition, themain advantage of relying on real variables is that we capture all type of recessions, nomatter if they have finan‐cial origin, energy prices, or any other cause. Whatever is the origin, if there is an effect on the economy, it has to be reflectedin national accounts type of variables. However, it is worth emphasizing that themain purpose of this analysis is to illustrate theadvantages of the proposed empirical framework with respect to alternative approaches. Therefore, improvements regardingthe most adequate selection of variables for each of the different regions under consideration are left for further research. Inall cases, the variables are seasonally adjusted, expressed in growth rates with respect to the previous period, and standardizedprior to enter the corresponding model.

3.2 THE CASE OF THE UNITED STATESDynamic factor models have been widely applied to the U.S. economy since it has been shown that they provide both accurateforecasts of GDP growth and inferences on the state of the economy in a real‐time environment. On the one hand, Giannone etal. (2008) and Banbura et al. (2012), among others, have relied on linear factor models, that allow for mixed frequency data, toprovide short‐term forecasts of real GDP. On the other hand, Chauvet (1998) employs single‐frequency nonlinear factormodels,where the factor is subject to regime changes, with aim of providing timely inferences on turning points. Moreover, Chauvetand Piger (2008) show that Markov‐switching dynamic factor models outperform alternative nonparametric methods wheninferring U.S. recessions as dated by the NBER.

We estimate, in a Bayesian fashion, a constant mean factor model by extending the approach in Chauvet and Piger (2008)to allow for mixed frequency data, and include quarterly real GDP growth to the set of monthly real activity indicators. This

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ASSESSING WEAKNESS ACROSS COUNTRIES

Figure 1United States

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Note: The blue solid line (right axis) plots the monthly probability of low real activity regime, Pr(st = 1), for the corresponding model, and the greybars (left axis) represent the real GDP quarterly growth.

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Table 1List of variables used for each economic region

ID Variable Frequency Sample

Australia

AU1 Real GDP Quarterly 1997:II – 2019:III

AU2 Industrial production index Monthly 1997:04 – 2019:08

AU3 Imports of goods and services Monthly 1997:04 – 2019:12

AU4 Exports of goods and services Monthly 1997:04 – 2019:12

AU5 Consumer sentiment index Monthly 1997:04 – 2020:02

Brazil

BR1 Real GDP Quarterly 1996:II – 2019:III

BR2 Industrial Production Index Monthly 2002:04 – 2019:12

BR3 Total production of vehicles Monthly 1996:04 – 2020:01

BR4 Retail trade: volume of sales Monthly 2000:04 – 2019:12

BR5 Merchandise exports Monthly 1996:04 – 2020:02

Canada

CA1 Real GDP Quarterly 1980:I – 2019:III

CA2 Monthly index of gross domestic product Monthly 1980:01 – 2019:12

CA3 Index of production of total industry Monthly 1980:01 – 2019:11

CA4 Exports: value goods in US Dollars Monthly 1980:01 – 2019:12

CA5 Imports: value goods in US Dollars Monthly 1980:01 – 2019:12

China

CN1 Real GDP Quarterly 1992:II – 2019:IV

CN2 PMI: Manufacturing (50+=Expansion) Monthly 2004:05 – 2020:02

CN3 Freight quantity: Imports and Exports (10,000 Metric Tons) Monthly 2012:02 – 2019:12

CN4 Exports: value goods in US dollars Monthly 1992:04 – 2019:12

CN5 Imports: value goods in US dollars Monthly 1992:04 – 2019:12

Euro Area

EA1 Real GDP Quarterly 1995:II – 2019:IV

EA2 Industrial production excluding construction Monthly 1995:01 – 2019:12

EA3 Intra‐euro area exports of goods Monthly 1999:02 – 2019:11

EA4 Extra‐euro area exports of goods Monthly 1999:02 – 2019:11

EA5 Industrial orders Monthly 1995:02 – 2020:02

India

IN1 Real GDP Quarterly 1996:III – 2019:III

IN2 Industrial production index: Electricity Monthly 2005:05 – 2019:12

IN3 Industrial production index excluding construction Monthly 1996:04 – 2019:12

IN4 Merchandise exports, f.o.b. in US dollars Monthly 1996:04 – 2020:01

IN5 Merchandise Imports, c.i.f. in US dollars Monthly 1996:04 – 2020:01

ID Variable Frequency Sample

Japan

JP1 Real GDP Quarterly 1993:II – 2019:III

JP2 Industrial production manufacturing index Monthly 1993:04 – 2020:01

JP3 Construction industry activity index Monthly 1993:04 – 2019:12

JP4 Exports of goods in Yens Monthly 1993:04 – 2020:01

JP5 Producer shipment durable consumer goods Monthly 1993:04 – 2019:12

Mexico

MX1 Real GDP Quarterly 1980:II – 2019:IV

MX2 Industrial production index Monthly 1980:04 – 2019:12

MX3 Motor vehicle production Monthly 1983:04 – 2020:01

MX4 Retail sales volume (Goods and Services) Monthly 1994:04 – 2019:12

MX5 Exports, f.o.b. in US dollars Monthly 1980:04 – 2020:01

Russia

RU1 Real GDP Quarterly 2003:II – 2019:III

RU2 Imports: value goods for the Russian Federation in US dollars Monthly 2003:04 – 2019:11

RU3 Exports: value goods for the Russian Federation in US dollars Monthly 2003:04 – 2019:11

RU4 Industrial Production Index Monthly 2006:04 – 2020:01

RU5 Passenger Car Sales (Imported Plus Domestic) Monthly 2006:04 – 2020:01

South Africa

SA1 Real GDP Quarterly 1980:II – 2019:III

SA2 Volume of production (manufacturing) Monthly 1980:04 – 2019:012

SA3 Merchandixe exports Monthly 1980:04 – 2020:01

SA4 New vehicules sold Monthly 1980:04 – 2019:12

SA5 Electricity production Monthly 1985:02 – 2019:12

United Kingdom

UK1 Real GDP Quarterly 1980:II – 2019:IV

UK2 Index of industrial production Monthly 1980:04 – 2019:12

UK3 Exports: value goods in Pounds Monthly 1980:04 – 2019:12

UK4 Passenger Car Registration Monthly 1980:04 – 2020:01

UK5 MFG Order Books Monthly 1980:04 – 2020:02

United States

US1 Real GDP Quarterly 1947:II – 2019:IV

US2 Industrial production index Monthly 1947:04 – 2020:01

US3 All employees, total nonfarm Monthly 1947:04 – 2020:01

US4 Real personal income excluding current transfer receipts Monthly 1959:04 – 2020:01

US5 Real manufacturing and trade industries sales Monthly 1967:04 – 2019:12

Note: The table reports the list of variables used in the models fitted to each economic region, along with frequency and coverage period.

extension constitutes our constant mean factor model. Our data starts in 1947 and ends in 2020, as shown in Table 1. Thissample period allows us to evaluate the performance of the model over the last eleven NBER recessionary episodes.⁷ Theestimated probability of low real activity regime, Pr(st = 1), is shown in Chart A of Figure 1, along with the data on GDPgrowth, for comparison purposes. Although the estimated probability reaches values close to one during eight of the elevenrecessionary episodes, the model does not provide a high recession probability for the three remaining recessions; 1969:12–1970:11, 1990:07–1991:03, and 2001:03–2001:11. This is because these three recessions seem to be either less severe, lesspersistent, or both, than the rest. Therefore, since the model assumes that the mean growth across all recessionary episodes isthe same based on the entire sample, the estimated mean of the factor during recessions is dominated by the eleven strongerandmore persistent recessions, and consequently, themodel fails to produce a high probability attained to the three remainingrecessionary episodes.

⁷ In particular, the last eleven U.S. recession, as defined by the NBER, are dated as follows: 1948:11–1949:10, 1953:07–1954:05, 1957:08–1958:04,1960:04–1961:02, 1969:12–1970:11, 1973:11–1975:03, 1980:01–1980:07, 1981:07–1982:11, 1990:07–1991:03, 2001:03–2001:11, 2007:12–2009:06.

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ASSESSING WEAKNESS ACROSS COUNTRIES

Figure 2United States: Real‐Time Performance

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Note: The figure plots the probabilities of low real activity computed with recursive out‐of‐sample exercises and with real‐time data, that is, byemploying the exact amount and type of information available at the time of the estimation. The blue line plots the probability computed with theconstant mean factor model, while the red line plots the probability computed with the time‐varying mean model. The dashed area correspond torecession periods, as dated by the NBER.

In order to account for the fact that some U.S. recessions could be substantially more severe than others, we estimate thetime‐varying mean factor model proposed in this paper, and plot its associated probability of low growth regime in Chart B ofFigure 1. The figure shows that the estimated probability reaches values close to one during all the NBER recessions, with noexceptions, outperforming the constantmean factormodel. The reason for the success of the proposed framework relies on thepremise that the growth rate during each recession is unique, in line with Eo and Kim (2016). However, unlike these authors,our approach is more parsimonious and simply assumes that recession means have two components. The first componentis deterministic and given by the estimated parameter 𝜇1, which provides an assessment about the average growth acrossall recessionary episodes in the sample. The second component is the random variable xt, which makes every recessionaryepisode unique, and which can be interpreted as deviations from the deterministic component 𝜇1. Hence, negative (positive)values of xt indicate weaker (stronger) growth than the average across all recessionary episodes, 𝜇1. Chart A of Figure 22, in theAppendix, plots the empirical posterior mean and median of the random variable xt over time for the case of U.S., illustratingthe need for an adjustment factor, especially, during the 1973:11–1975:03 and 2007:12–2009:06 recessions.

Next, we turn to evaluate the performance of the two competing factor models in a real‐time environment. First, we estimateboth models with revised data from 1947:02–1989:12. Second, we perform recursive estimations with unrevised (that is, real‐time) vintages of data, from 1990:01 until 2019:08, adding one month of information at every time. The real‐time vintages forthe variables in themodel were retrieved from the Archival of the FRED economic datase.⁸ The estimated real‐time probabilitiesof low activity regime associated to both models are plotted in Figure 2. Several features deserve to be commented. First, thereal‐time inferences, obtained with the constant mean factor model, are able to provide a better track of the last three NBERrecessions than the full‐sample (1946–2019) inferences obtainedwith the samemodel. This is because inferences on a recessionin the present are not ‘contaminated’ by any information (for example, deepness) associated to recessions in the future. Second,the real‐time probabilities coming from the constant mean factor model accurately infer the end of the recessions. However,they fail to provide a ‘timely alarm’ for the beginning of the recessionary episodes. This feature is aligned with the results inCamacho et al. (2018), who rely on a classical estimation approach. Third, the real‐time probabilities obtained with the time‐varying mean factor model tend to quickly rise to values close to one at the beginning of the recessions, and drop to values

⁸ https://alfred.stlouisfed.org/.

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close to zero at their end. This result emphasizes the usefulness of the proposed approach in providing timely updates of thestate of the economy in the real‐time environment that policy makers and investors face.⁹

3.3 THE CASE OF THE EURO AREA¹⁰Dynamic factor models have been also applied for nowcasting purposes in the euro area, either from a single economy per‐spective (Banbura et al., 2011) or for individual member countries (Runstler et al., 2009). More related to our study, Camachoet al. (2014) provide a nonlinear extension, which allows the mean factor to switch between regimes of high and low growth,with the aim of inferring the state of the euro area economy in real‐time.

Producing accurate inferences on the state of the euro area economy is associated to an important challenge regarding thedifferences in the definition of a low activity regime. According to the Euro Area Business Cycle Dating Committee of theCentre for Economic Policy Research (CEPR), since the introduction of the euro, there have been two episodes that can betechnically categorized as recessions; 2009:03–2010:10 and 2012:11–2015:10. While during the first period the average growthwas ‐0.9%, in the second one the average growth reached only to ‐0.2%. In other words, the 2009:03–2010:10 recession wasabout 4.5 times more severe than the 2012:11–2015:10 recession. As we illustrate below, this particular feature has importantimplications for the performance of nonlinear factor models in defining regimes of low and high growth.

We estimate a constant mean factor model for the Euro Area, with the corresponding data reported in Table 1. For the sakeof space, the estimated probability of low growth regime is plotted in Chart A of Figure 10 in the Online Appendix. The modelattains a high probability of low growth only to the slowdown associated to the Great Recession, and fails to detect otherperiods of negative (2012–2013) or weak (2001–2004) output growth. This is because during the Great Recession, the EuroArea exhibited an unprecedented deterioration in real activity, which fully dominates the estimated low mean growth. Hence,similarly to the case of the U.S., the assumption that all recession are alike turns to be detrimental to accurately infer regimes oflow real activity. Next, we evaluate the performance of the proposed time‐varying mean factor model, and plot the probabilityof low growth in Chart B of Figure 10. The estimates show that when allowing for heterogeneous means, the model is able toattain a high probability of low growth to all the episodes associated to either negative or weak output growth in the sample.¹¹

3.4 OTHER ADVANCED ECONOMIESAs just pointed, factor models subject to regime changes have been successfully used for the U.S. and Euro Area economies.Here, we evaluate how this type of models perform when facing data of a different nature associated to other advancedeconomies.

AUSTRALIAThe growth of real GDP in Australia has generally remained at positive values since the mid 1990s, with only a few exceptionalquarters of negative growth. This feature entails a significant challenge in assessing the state of its economy based on regime‐switchingmodels previously used in the literature. To illustrate this point, we estimate the constant mean factor model with theAustralian data reported in Table 1. The associated probability of low real activity is plotted in Chart A of Figure 11, reportingvalues between 0.2 and 0.7, and therefore, showing that this model is not able to clearly identify the presence of more thanone regime associated to the underlying data. Next, we estimate the time‐varying mean factor model, and plot the associatedprobability of low real activity in Chart B of Figure 11. The estimated inferences show the time‐varying meanmodel provides aclearer identification of a low activity regime, occurred in 2009, than the constant meanmodel.

⁹ In addition, we evaluate the ability of the two alternative models, constant mean and time‐varying mean, to infer U.S. recessions, as dated by theNBER. To do so, we employ the Receiver Operating Characteristics (ROC) curve and show that the time‐varying mean outperforms the constant meanboth in full‐sample and, especially, in real‐time. See charts A and B of Figure 21 of the Online Appendix. For more details about the ROC curve, seeBerge and Jordà (2011).

¹⁰ THE EURO AREA MODEL HAS BEEN ESTIMATED IN THE CONTEXT OF THE TASK FORCE OF THE EUROSYSTEM ON NONLINEAR TOOLS.¹¹ In Chart B of Figure 22 of the Online Appendix, it can be seen that the adjustment of the mean growth during the period associated to the GreatRecession was sizeable in comparison to the other times of weak activity, due to its severity. Also, after computing the ROC curve for the presentcase, Chart C of Figure 21 of the Online Appendix shows that the time‐varying meanmodel outperforms the constant‐meanmodel in inferring EuroArea recessions, as dated by the CEPR. For more information about the ROC curve, see Berge and Jordà (2011).

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ASSESSING WEAKNESS ACROSS COUNTRIES

CANADAInferring the state of the Canadian economy is expected to face similar challenges to the case of the U.S., given their close ties.¹²We estimate the constant mean factor model with the Canadian data reported in Table 1. The estimated probability of low realactivity is plotted in Chart A of Figure 12. Notice how the downturn in real activity associated to the Great Recession dominatesthe regime inferences, since the model only provides high probability to that specific period. However, as the figure shows,GDP growth has exhibited several additional episodes of weakness, which are totally missed due to the assumption that allrecessions are alike. Next, we estimate the time‐varying factor model with Canadian data, and plot the associated probabilityof low real activity in Chart B of Figure 12. The changes with respect to the Chart A are apparent. The time‐varying factormodel is able to produce monthly probabilities that match fairly well with episodes of either weak or negative GDP growth,constituting a more reliable tool for assessing the state of the economy than the constant mean factor model.

JAPANNext, we focus on the economy of Japan, whose real GDPwas significantly contracted during the Great Recession in comparisonwith other temporary downturns occurred since the mid 1990s. We estimate the constant mean factor model with Japanesedata, and plot the associated probability of low activity in Chart A of Figure 13. The estimated inferences identify two periodsof low real activity, corresponding to the downturns of 2009 and 2012. Instead, when relying on the time‐varying mean factormodel, additional episodes of low real activity are identified, as shown in Chart B of Figure 13. Notice that these additionalepisodes are associated to smaller, but still negative, declines in output growth than the ones occurred in 2009 and 2012. Thisillustrates the ability of our framework to identify economic downturns associated to heterogeneous deepness.

UNITED KINGDOMSince the ‘Brexit’ referendum of 2016, the U.K. economy has been subject to a high level of uncertainty, which has increasedthe interest in inferring a potential upcoming recession. However, recent GDP downturns, occurred since the mid 1980s, havebeen in general substantially smaller than the contraction exhibited during the Great Recession. This feature could preclude amodel to accurately infer the next recession if it is of a smaller magnitude than the one observed in 2008‐2009. We estimatethe constant mean factor model for U.K. data, and plot the corresponding probability of low activity in Chart A of Figure 14.As expected, this model is only able to infer the significant downturn associated to the 2008‐2009 recession, missing otherperiods of negative GDP growth. Next, we estimate the time‐varying mean factor model, and plot the associated probability oflow activity in Chart B of Figure 14. The estimates attain a relatively high probability to additional periods associated to eitherlow or negative output growth.

3.5 EMERGING MARKETSThere is a flourishing literature focused on employing dynamic factor models to provide nowcasts of activity in emerging mar‐kets. This framework has been applied for Brazil (Bragoli et al., 2015), Mexico (Corona et al., 2017), Russia (Porshakov etal., 2016), India (Bragoli and Fosten, 2017), and China (Yiu and Chow, 2010). Also, some works focus comparing the perfor‐mance of factor models in more than one country. This is the case of Cepni et al. (2019a) and Cepni et al. (2019b) who compareaddress the cases of Brazil, Indonesia, Mexico, South Africa, and Turkey. Also, Dahlhaus et al. (2015) encompass a wider sampleof countries, by also including Russia and China into the analysis. The overall message of these works, is that factor models alsotend to be successful in providing accurate short‐term forecast of output growth for this type of countries. However, the liter‐ature on assessing the state of the emerging economies is rather scarce. In this section, we employ the proposed frameworkto provide timely inferences on low real activity regimes for six of the largest emerging markets; Brazil, Mexico, Russia, India,China and South Africa.

For the case of Brazil, Chauvet (2001) proposes the use of a single‐frequency factor model subject to regime‐changes with theaim of inferring recessions, with data prior to 2000, finding several episodes of low real activity. We fit the constant mean factor

¹² In the previous work by Chernis and Sekkel (2017), dynamic factormodels are employed to produce nowcasts of the Canadian GDP growth. Although,to the best of our knowledge, ours is the first study that uses nonlinear factor models to infer its regimes of high and low real activity.

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model to Brazilian data reported in Table 1, with data starting in 1996 until the present time. The estimated probabilities of lowactivity are plotted in Chart A of Figure 15, showing that the inferences are dominated by the acute downturn associated to theGreat Recession, reaching a quarterly growth of around ‐4 percent. Consequently, the model is not able to categorize as lowactivity regimes to several episodes of negative, or even close to zero, output growth. Instead, the probabilities obtained withthe time‐varying mean factor model provides a more accurate assessment of weak activity periods of the Brazilian economy,such as the one between 2014 and 2015. Also, the correction in the recession mean growth needed to improve the inferenceexhibits different magnitudes and directions, as can be seen in Chart D of Figure 22, indicating a strong idiosyncracy of realactivity downturns in Brazil.

We also apply the proposed framework to the cases ofMexico, Russia, India, China and South Africa and plot the correspondingprobabilities of low activity regime in figures, 16, 17, 18, 19, and 20, respectively. In all five cases, the results indicate that thetime‐varyingmean factormodel outperforms the constantmeanmodel in that the former is able tomake a better track of weakreal activity periods in emerging markets, which are aligned with the downturns in GDP growth. All these results emphasizethe flexibility of our approach in adapting to economies with substantially diverse types of activity dynamics to provide a robusttracking of its strength.

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4 Assessing Global Weakness

The Great Recession carried out severe and long‐lasting negative consequences for world economy. Recently, the IMF hascalled the attention for synchronized global slowdown in economic activity since many of the largest economies are exhibitingdecelerations in their output growth pace (IMF, 2019) and different institutions consider that the coronavirus crisis has putthe economy at the verge of a recession. Therefore, inferring the state of the world economy on a timely basis is of greatimportance. In a recent study, Ferrara and Marsilli (2018) propose a framework to produce short‐term forecasts of the globaleconomic growth by also using factor models, which are fitted to economic indicators from different advanced and emergingeconomies. Although, the authors rely on a linear approach, since they only focus on nowcasting, and not on inferring the stateof the global economy.

The framework proposed in this paper has been shown to provide accurate and timely assessment about the weakness ofthe economic activity in some of the largest advanced and emerging economies. In this section, we propose a simple way tocombine all those assessments into a single index that proxies the weakness of the world economy and that can be updatablein real‐time. The purpose of the proposed index is threefold; (i) measuring the weakness of the world economy, (ii) identifyingthemain underlying sources of global weakness, and (iii) providing risk assessments of global downturns associated to differentintensities.

4.1 DYNAMICS AND SOURCES OF WEAKNESSTo provide a comprehensive view of the evolving heterogeneity of the economic weakness across different regions, the infor‐mation on their corresponding probabilities of low activity are plotted in world maps for selected periods. Chart A of Figure3 shows the global situation for 2008:11, around the middle of the last global recession. As expected, all the regions underconsideration reported a probability of low activity close to one, exhibiting a high synchronization in a ‘bad’ state. Instead,Chart B plots the global situation for 2010:01, the beginning of the year with the strongest annual world GDP growth sincethe Great Recession (5.4 percent). This is also a very homogeneous period, however, in this case all the regions reported aprobability of low activity close to zero, exhibiting a high synchronization in a ‘good’ state. However, not all time periods areaccompanied by such a high degree of synchronicity. In general, there is substantial heterogeneity regarding the state of theeconomy across regions. An example of that is shown in Chart C, which plots the situation for the last period in our sample,2020:02. Notice that some regions, such as the Euro Area, currently show a position substantially weaker than others that facea stronger activity. However, the most striking feature of this map is the high weakness of the Chinese economy, induced bythe coronavirus crisis. An advantage of this map is that it can be updated every day that a new datum, associated to any of theregions under consideration, is published. These updates would help to reassess the weakness of a given economy in globalcomparative terms. The entire sequence of maps for the period 2003:04‐2020:02 is available online.¹³

We are interested in providing a single statistic that summarizes the state of the global economy, and that additionally can be(i) updated in real‐time, (ii) decomposed into its regional contributions, (iii) useful to quantify risks, and (iv) simple to interpret.Therefore, we proposed the Global Weakness Index (GWI) which consists on a weighted average between the probabilities oflow activity associated to each of the K regions under consideration. Since the models are estimated in a Bayesian fashion, weare able to reproduce many replications associated to the realization of a low activity regime for each region, that is, s(l)𝜅,t, andfor 𝜅 = 1, ..., K, and l = 1, ..., L, where K is the number of countries and L is the number of draws. L should be large enough toensure convergence in the associated posterior density. Hence, the l‐th replication of the GWI is given by,

GWI(l)t =K

𝜅=1

𝜔𝜅,ts(l)𝜅,t, (8)

where𝜔𝜅,t denotes the time‐varying weights associated to each region based on its evolving economic size relative to the worldGDP. The collection of all the replications {GWI(l)t }l=L

l=1 constitutes the simulated density of the index at time t, from which point

¹³ It can be found at https://sites.google.com/site/daniloleivaleon/global_weakness.

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Figure 3Weakness Across Countries

(a) Middle of the Last Global Recession

(b) Beginning of the Last Global Expansion

(c) End of Sample

Note: The heatmap of the world plots the overall pattern of the probabilities of low economic activity regime Pr(st = 1) for specific periods. Thedarker (lighter) the area, the higher (lower) the probability of low economic activity. The animated sequence of world heatmaps, from 2003:04 until2020:02, can be found at: https://sites.google.com/site/daniloleivaleon/global_weakness.

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ASSESSING GLOBAL WEAKNESS

estimates along with any percentile, to measure its uncertainty, can be easily computed. Notice that, by construction, theGWI is bounded between zero and one. That is, the index would take the upper (lower) bound value of one (zero) when theprobabilities of weak activity associated to all the regions under consideration are equal to one (zero), implying the highest(lowest) degree of global weakness.

In Chart A of Figure 4, we plot the Global Weakness Index, along with the world GDP growth, for comparison purposes.¹⁴There are several important features. First, notice that the GWI starts to rapidly increase during 2008, reaching a value of0.8 in October of the same year, indicating clear signs of a global contraction. Second, as the GWI increases during 2008, thecorresponding credible set shrinks, indicating a high synchronization in a ‘bad’ state.¹⁵ Third, the GWI anticipates the exit ofthe last global recession by dropping to values close to zero around July 2009. Fourth, the index also detects two episodesof moderate weakness for the world economy. The first one took place between the late 2015 and the early 2016, periodof slower growth in emerging markets and gradual pickup in advanced economies, as characterized by the IMF. The secondepisode of moderate weakness is the present one. There is a moderate increase in weakness since 2018 that skyrockets withthe information about coronavirus crisis, taking the index to its largest value since the beginning of the Great Recession, in theearly 2008.

The full‐sample estimates of the GWI, shown in Chart A of Figure 4, are of high importance in order to compare the currentstate of the world economy with respect to the past, based on all available information. However, it is also relevant addressingwhether the GWI index is able to provide accurate real‐time assessments of the state of the world economy, that is, by usingonly the available information at the moment of the estimation. In doing so, we recursively estimate all the models associatedto the 12 economic regions under consideration by including onemonth of information at a time. Our estimations of themodelsare based on expanding windows, staring at the beginning of the sample for each region, as reported in Table 1, and ending inperiods from 2007:01 until 2020:02.¹⁶ Next, we compute the GWI associated the each of the recursive estimations, and collectthe last available estimates for each vintage of data. The sequence of these collected GWI estimates, shown in Chart B of Figure4, represent the assessments that our proposed framework provides about the state of the world economy at every period,constructed by using only the available information at that point in time. Notice that the real‐time GWI index resembles fairlywell the full‐sample estimates, indicating a robust performance in inferring regimes of low activity in a timely fashion. Thisfeature can be particularly observed at the beginning of the Great Recession, where the GWI provided a strong real‐time alert,taking values higher than 0.5 during the first months of 2008.

to the different regions (in Table 1) is released the GWI can be updated. To the best of our knowledge there is no other model‐based index that provides updates on the state of the global economy with such a high frequency. Yet, there is a non‐model‐based index that is able to proxy the overall ‘sentiment’ of agents about the likelihood of an upcoming global recession. Thisindex can be computed by counting the number of web searches of the words “Global Recession” performed by internet userfrom all over the world, and therefore, can also be updated at the daily frequency. Figure 5 plots the web search index, whichis associated to agents’ inference about a global recession, along with the GWI, which is based on information of real economicactivity. The figure shows that despite both indexes pick up around the last global recession, as dated by Kose et al. (2020), theGWI substantially leads the web search index. In particular, the web search index increases once the last global recession wasalmost in the middle of its duration, and substantially declined (below levels of 0.2) about a year after the beginning of the lastglobal expansion. Also, while the web search index has remained relatively unchanged since 2010, the GWI has indicated signsof moderate economic global weakness at different points in time. These results provide comparative evidence in favor of thetimeliness of the GWI to infer global downturns.

The contributions of advanced and emerging economies to the total world GDP have exhibited sizable changes over the lastyears. While, for example, the shares of U.S. and Euro Area have declined, the relative economic importance of China andIndia has surged with time. This evolving composition plays a important role when disentangling the main sources of weaknessin global activity. As defined in Equation (8), the GWI can be easily decomposed into the contributions associated to eacheconomic region. Each of these contributions has two components. First, the likelihood of falling in a low activity regime,E[s𝜅,t], and second, the economic importance of such likelihood for the world GDP,𝜔𝜅,t. The product of these two componentshelps to identify the main contributors of the degree of weakness of the global economy, either it is high or low.

¹⁴ The world GDP is taken from the IMF database.¹⁵ The credible set of the GWI is define by the percentiles 16th and 84th of the simulated distribution.

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Figure 4Global Weakness Index

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Note: The figure shows the Global Weakness Index, which is constructed as a weighted average of the probabilities of low economic activity regimeacross countries. The weights are given by the size of the corresponding economies. The red area represents the credible set based on the percentiles16th and 84th of the posterior distribution. The dashed blue line plots the world real GDP growth based on the estimates provided by the IMF, forcomparison purposes. Chart A plots the estimates based on the entire available sample for each world regions. Chart B plots the real‐time estimates,which are computed with the available set of information for each world region at the moment of the estimation. The last observation correspondsto February 2020.

In Figure 6, we plot the GWI along with its corresponding regional contributions, which are normalized to sum up to one forevery time period for ease of interpretation. The figure shows the substantial importance of the U.S. in the deterioration ofglobal economic conditions at the beginning of the last global recession, around the end of 2007. However, in 2008 the relativecontribution all the other regions, in particular, advanced economies, starts to rise, yielding amore uniform composition during

¹⁶ Due to the complexity of the international environment we are dealing with, there are data availability constraints, and therefore, our estimationsdo not take into account data revisions of the associated variables.

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ASSESSING GLOBAL WEAKNESS

Figure 5Timeliness of the Global Weakness Index

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020

2008-2009 Global Recession Web search on "Global Recession" Global Weakness Index

Note: The figure shows the real‐time Global Weakness Index (red solid line) and a monthly index numbers that represent web search interest for theterm “global recession” (blue dashed line). The web search index is constructed relative to its highest point, with a peak popularity for the term thatequals to 1. The information to construct the web search index is obtained from Google Trends. The grey bar represents the period of the last globalrecession (2008:III‐2009:I), as dated in Kose at al. (2020). The last observation corresponds to February 2020.

the middle that global recession, by 2009. This pattern is consistent with a sequential propagation of recessionary shocks. Thesubsequent recovery, which started around late 2009, and posterior expansion, since 2010, was instead characterized by adegree of global weakness close to zero, where the main contributors were emerging markets.

This buoyant global economic stance lasted for about a year. Since 2011 the GWI experienced slight, but sustained, increases,drivenby the Japanese economydue to a sequenceof unexpected events, such as, an earthquake, a tsunami, and the Fukushima’snuclear crisis. In 2012 the GWI increased another point, reaching to 0.3, although with a different composition. This time, themain contributor was the Euro Area due to the sovereign debt crisis. During 2015, there was another slight increase in theGWI, which can be mainly attributed to Russia due to the international economic sanctions and oil shocks that were highlydetrimental for its financial conditions. Lastly, the decomposition also illustrates the large impact that the recent deteriorationin Chinese economic conditions have had on the increasing weakness of the global economy. Overall, the GWI is shown toprovide a narrative of events that have led to either slight or substantial increases in the weakness of the global economy.

4.2 MONITORING GLOBAL RISKS

In a recent work, Adrian et al. (2019) illustrate the importance of evaluating downside risks of U.S. output growth associated totighter financial conditions. The authors rely on quantile regressions tomodel the conditional predictive densities of real activity.Thereafter, this methodology has also been applied to a wide variety of individual countries for macroeconomic surveillancepurposes (Prasad et al., 2019). Assessing the likelihood of upcoming extreme events, or “macroeconomic disasters”, given thecurrent conditions, is crucial for policymakers. Although, this task becomes more challenging when the target to be monitoredis the world economy, as a whole. This is due to the lack of, relatively, high frequency economic data at the world level.

In this section, we employ the proposed GWI as a tool for monitoring risks at the global level and in real‐time. Also, due tothe way in which the GWI is constructed, it is straightforward to provide timely assessments about two important features ofthe world economy. First, the evaluation of risks associated to a state of low global activity. Second, the regional contributions

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MAGYAR NEMZETI BANK

Figure 6Historical Decomposition of Global Weakness

2004 2006 2008 2010 2012 2014 2016 2018 20200

0.5

1

South Africa Mexico Australia Russia Canada Brazil U.K. India Japan China Euro Area U.S.

2004 2006 2008 2010 2012 2014 2016 2018 20200

0.5

1

Note: The top chart plots normalized contributions of each economic region to the GlobalWeakness Index. The contribution of each region, at a giventime, is defined as the product between the associated (i) probability of low economic activity, and (ii) the weight of the economy, which is definedby the its relative size in terms of GDP, see Equation (8). The contributions are normalized to sum up to one at every period. The bottom chart plotsthe full‐sample estimates of the Global Weakness Index for reference purposes. The last observation corresponds to February 2020.

associated to such risks. Figure 7 shows the weakness stance of the global economy during four selected periods. First, ChartA shows the situation at the beginning of the last global recession (2008M03), where most of the mass of the GWI distributionstarted to displace towards the right, suggesting an upcoming weakening of the global economy. Also, corresponding theradar chart indicates that most of such a weakening was originated in the U.S. economy. Chart B shows the same type ofinformation, but for a period around the middle of the last global recession (2009M01), where most of the mass of the GWIdistribution was compressed at values close to one, indicating a severe global downturn. The associated radar chart revealsthat the GWI composition in this period was relatively homogeneous between the largest world economies, that is, U.S., theEuro Area and China, due to the international propagation of contractionary shocks. Chart C shows the global risk assessmentduring a period of buoyant recovery (2009M12), which is reflected in GWI distribution, concentrated in values close to zero.Interestingly, not only the U.S. and the Euro Area were the drivers of this trend, but also the economies of Japan and Russiasignificantly contributed. Lastly, Chart D plots the current situation (2020M02), which suggests increasing risks of high level ofglobal weakness, with China being the main driver due to the coronavirus outbreak.

The information contained in the time‐varying GWI distributions, plotted in Figure 7, can be also useful to quantify the proba‐bility associated to specific scenarios of weakness in global activity. For example, if one is interested in assessing the probabilitythat the weakness of the global economy would exceed a low level. In this case, we define a low level by a value of GWIt = 0.3,and compute the associated cumulative density. Similarly, we compute the probability the weakness of the global economywould exceed a medium, high, and very high level, defined by values of 0.5, 0.7, and 0.9, of the GWI, respectively. These valuesare used just as reference and can be changed based on the judgement of the policy markers or pundits.

Chart A of Figure 8 plots the time‐varying probabilities associated to the scenarios in which the global weakness is higher thanthe selected thresholds. As expected, the larger is theGWI threshold, the less likely tends to be such an scenario. The probabilityof exceeding a low global weakness level varies substantially over time and, with the exception of the periods around 2010 and2017, it tends to be a highly likely scenario. Instead, the probability of exceeding a high level of global weakness tends to berelatively low over time, with the exceptions of the period between 2008 and 2009. Lastly, the probability of a ‘catastrophe’,that is, when the global weakness exceeds a very high level reaches to values close to one at the beginning of 2009, and haveremained close to zero during the rest of the sample.

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ASSESSING GLOBAL WEAKNESS

Figure 7Real‐Time Monitoring of Global Risks

2008M3: The coming of the storm 2009M01: Middle of the last global recession

2009M12: A buoyant global economy 2020M02: Current situation

Note: The figure contains four charts associated to different time periods of interest. Each chart contains two sub‐charts. The sub‐charts on the leftprovide information on the posterior distribution of the (real‐time) GlobalWeakness Index, where the gray bars show the histograms, red dashed linesindicate the kernel densities, and blue solid lines make reference to the median of the corresponding posterior distributions. The radar sub‐charts onthe right show information about the relative contribution of each economic region to the Global Weakness Index. The contributions are normalizedto sum up to one.

To provide a broader perspective of these probabilistic assessments, in Chart B of Figure 8, we plot the same information,but allowing for a continuum of the GWI thresholds, information that can be interpreted as the density function of the globaleconomic weakness in every period of time. The plotted surface illustrates the constantly changing nature of global economicrisks. It is worth emphasizing that all these probabilistic assessments are constructed onlywith the available (revised) data at thetime of the estimation. To illustrate the readiness of the GWI, we zoom into the period before and during the Great Recession.That is, we slice the surface of global weakness at selected time periods between 2007 and 2009 and plot this informationin Figure 9, showing that as the world economy enters the last global recession, the curves relating degree of weakness withprobability of occurrence change from concave, in February 2007, to convex, in January 2009. Therefore, this displacements ofthe curves indicate a progressive rise in global economic risks. These results illustrate that GWI is able to provide a robust andtimely evaluation of economic risks associated to the world economy.

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MAGYAR NEMZETI BANK

Figure 8Risk Assessment of Global Weakness

(a) Selected Degrees of Weakness

2008 2010 2012 2014 2016 2018 20200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Low: Pr(GWIt) > 0.3

Medium: Pr(GWIt) > 0.5

High: Pr(GWIt) > 0.7

Very High: Pr(GWIt) > 0.9

(b) All Degrees of Weakness

Degree of Weakn

ess

Time

0

0.2

0.40

2008 0.620102012

0.820142016

2018 12020

0.5

Pro

babi

lity

of O

ccur

renc

e

1

Note: Chart A plots the time‐varying probability associated to a scenario in which the GWI reaches values higher than a given threshold. The selectedthresholds are 0.3, 0.5, 0.7 and 0.9. Chart B plots the time‐varying probability associated to a scenario in which the GWI reaches values higher thana any value between 0 and 1. These probabilities are computed based on the posterior density of the Global Weakness Index. The last observationcorresponds to February 2020.

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ASSESSING GLOBAL WEAKNESS

Figure 9Risk Assessment of Global Weakness Before and During the Great Recession

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Pro

bab

ilit

y o

f O

ccu

rren

ce

Degree of Weakness

2007-02 2007-04 2008-02 2008-04 2008-06 2008-09 2008-11 2009-01

Note: The figure plots the curves that relate level of global weakness with probabilities of occurrence for selected time periods before and during thelast global recession. The definition of the selected time periods is shown in the bottom legend.

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5 Conclusions

Motivated by recent events that signal a potentially global synchronized slowdown, we propose an econometric framework ableto provide timely and accurate real‐time inferences about the state of the world economy. We introduce a mixed frequencydynamic factor model that allows for heterogeneous deepness across recessionary episodes. The proposed model is fitted totwelve of the world’s largest economic regions. Our estimates show that allowing for heterogeneous recessions turns to becrucial in accurately inferring periods of weak real activity growth associated with both advanced and emerging economies,outperforming frameworks previously proposed in the literature. Next, the estimated regional inferences are summarized intoa Global Weakness Index (GWI), which is able to provide daily real‐time updates of the global economy strength, its underlyingsources, and risk assessments, as new information across the regions is released.

The proposed framework for monitoring the state of the world economy can be extended in different ways. For example,by determining the most adequate set of economic indicators associated to each region in order to maximize the accuracywhen inferring region‐specific recessions. Also, a wider range of regions can be incorporated in the construction of the globalweakness index with the aim of sharpen the accuracy when inferring world recessions.

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References

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MAGYAR NEMZETI BANK

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[36] Porshakov, A., A. Ponomarenko, and A. Sinyakov (2016). Nowcasting and short‐term forecasting of Russian GDP with adynamic factor model. Journal of the New Economic Association 30(2):60–76.

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[38] Runstler, G., K. Barhoumi, S. Benk, R. Cristadoro, A. Den Reijer, A. Jakaitiene, P. Jelonek, A. Rua, K. Ruth, and C. VanNieuwenhuyze (2009). Short‐term forecasting of GDP using large datasets: a pseudo real‐time forecast evaluation exer‐cise. Journal of Forecasting 28(7):595–611.

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Appendix A Model and Estimation

A.1 FULL SPECIFICATION OF THE MODEL

The common factor ft affects all the observed variables; it evolves according to the following rule:

ft = 𝜇0(1− st) + 𝜇1st + stxt + ef,t, ef,t ∼ 𝒩(0, 𝜎2f ). (A.1)

The indicator st ∈ {0, 1} equals one whenever there is an abnormal episode; it is a two‐state Markov‐switching process whoseevolution is summarized by p = Pr(st = 1|st−1 = 1) and q = Pr(st = 0|st−1 = 0). In case of an abnormal episode (st = 1), thecommon factor is augmented by an unobserved variable xt, which evolves as follows:

xt = stxt−1 + (1− st)vt, vt ∼ 𝒩(0, 𝜎2v ). (A.2)

Whenever there is an abnormal episode, the common factor is augmented by xt, which takes a random value at the beginningof the episode and then remains constant for the duration of the episode.

Each monthly variable is assumed to be a combination of the common factor ft and an individual component ui,t:

ymi,t = 𝛾ift + ui,t. (A.3)

Each individual component is assumed to have P lags:

ui,t = 𝜓i,1ui,t−1 + ... + 𝜓i,Pui,t−P + ei,t, ei,t ∼ 𝒩(0, 𝜎2i ). (A.4)

In addition, following Mariano and Murasawa (2003), the growth rate of a variable observed with quarterly frequency (such asthe GDP) can be expressed as a combination of its monthly unobserved growth rates as follows:

yqj,t =13ymj,t +

23ymj,t−1 + ymj,t−2 +

23ymj,t−3 +

13ymj,t−4. (A.5)

In turn, the monthly growth rates have the same decomposition as described in equation (A.3), so that we can express eachquarterly growth rate as a combination of the common factor and the individual component, as follows:

yqj,t =13𝛾jft +

23𝛾jft−1 + 𝛾jft−2 +

23𝛾jft−3 +

13𝛾jft−4 +

13uj,t +

23uj,t−1 + uj,t−2 +

23uj,t−3 +

13uj,t−4. (A.6)

As for the individual components of the quarterly series, they have the same structure as those described by equation (A.4) forthe monthly series.

Let there be Q quarterly observable variables andMmonthly observable variables, and let us summarize them by vector yt =[yq1,t, ..., y

qQ,t, ymQ+1,t, ..., ymQ+M,t]′. To summarize equations (A.3)–(A.6) into an observation equation, we need to define the vector

of unobserved states zt = [ft, ..., ft−4, u1,t, ..., u1,t−4, ..., uQ,t, ..., uQ,t−4, uQ+1,t, ..., uQ+1,t−P+1, ..., uQ+M,t, ..., uQ+M,t−P+1]′. As we can see,the vector zt combines the common factor with lags up to 4 and individual components of the quarterly variables with lagsup to 4 in order to account for the representation of quarterly variables according to equation (A.6); it also includes individualcomponents of the monthly variables with lags up to P− 1.¹⁷ Then, assuming that all the variables are observed in period t, wecan formulate the observation equation:

yt = Hzt + 𝜂t, 𝜂t ∼ 𝒩(0,R) (A.7)

¹⁷ Because the lags for quarterly variables’ individual components are capped at 4, this specification effectively restricts P to be no greater then 5. Thisrestriction can easily be relaxed.

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MAGYAR NEMZETI BANK

where 𝜂t is a vector of measurement errors, and (Q+M) × (5+ 5Q+ PM)matrix H is the following:

H =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

𝛾1 × HQ HQ ⋯ 0 0 ⋯ 0

⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋮

𝛾Q × HQ 0 ⋯ HQ 0 ⋯ 0

𝛾Q+1 × H5M 0 ⋯ 0 HP

M ⋯ 0

⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋮

𝛾Q+M × H5M 0 ⋯ 0 0 ⋯ HP

M

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (A.8)

HQ = 13

23

1 23

13, H5

M = 1 0 0 0 0 , HPM = 1 0 ... 0 .

Note that the size of the matrix HPM is 1 × P, and the only non‐zero element is the first one. More generally, in periods when

some of the observations are missing, the observation equation can be cast without the rows that correspond to the missingobservations:

y∗t = Htzt + 𝜂∗t , 𝜂∗t ∼ 𝒩(0,Rt) (A.9)

where Ht is obtained by taking H and eliminating the columns that correspond to the missing variables, and the matrix Rt isobtained by eliminating the corresponding rows and columns from matrix R.

Next, let us define the dynamics of the unobserved state zt:

zt =

⎡⎢⎢⎢⎢⎢⎢⎣

st𝜇0 + (1− st)𝜇1 + stxt

0

0

⎤⎥⎥⎥⎥⎥⎥⎦

+ Fzt−1 + 𝜀t, 𝜀t ∼ 𝒩(0,Q), (A.10)

In this equation, F is a (5+ 5Q+ PM) × (5+ 5Q+ PM)matrix, which can be compactly expressed as follows:

F =

⎡⎢⎢⎢⎢⎢⎢⎣

F0 0 ⋯ 0

0 Ψ1 ⋯ 0

⋮ ⋮ ⋱ ⋮

0 0 ⋯ ΨM+Q

⎤⎥⎥⎥⎥⎥⎥⎦

, where F0 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

As for thematrixΨi, it is a 5×5matrix for quarterly series (i = 1, ...,Q), since there are four lags ofmonthly individual componentsfor each quarterly series in the state vector zt:

Ψi =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

𝜓i,1 𝜓i,2 𝜓i,3 𝜓i,4 𝜓i,5

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

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APPENDIX A MODEL AND ESTIMATION

However, the coefficients 𝜓i,p are non‐zero only for p < P, where P is the number of specified lags.¹⁸ In case of monthly series(i = Q+ 1, ...,Q+M), the matrixΨi is P × P.

Vector 𝜀t contains shocks to the common factor and each individual component:

𝜀t = [ef,t, 0, 0, 0, 0], [e1,t, 0, 0, 0, 0], ..., [eM+Q,t, 0, 0, 0, 0]′. (A.11)

Correspondingly, the matrix Q is a diagonal matrix, such that

diag(Q) = [𝜎2f , 0, 0, 0, 0], [𝜎2

1 , 0, 0, 0, 0], ..., [𝜎2M+Q, 0, 0, 0, 0] . (A.12)

Note that the dynamics equation (A.10) contains an extra vector that depends on the state indicator st and the latent variablext—this does not complicate the application of the Kalman filter, since the Bayesian estimation that we use takes turns tosimulate {zt}Tt=1, then {xt}Tt=1, and then {st}Tt=1. For example, during the step that simulates {zt}Tt=0, the value st𝜇0+(1− st)𝜇1+stxt is fixed and therefore treated as a constant.

Equations (A.2), (A.7)–(A.12) summarize the model that we estimate, and the model is summarized by parameter vector 𝜃:

𝜃 = p, q, 𝜇0, 𝜇1, 𝜎v, 𝜎f, 𝛾1, ..., 𝛾Q+M, {𝜓1,1, ..., 𝜓1,P}, ..., {𝜓Q+M,1, ..., 𝜓Q+M,P}, 𝜎1, ..., 𝜎Q+M .

A.2 BAYESIAN ESTIMATION

Let Y = {yt}Tt=0 be the observed data that consists of quarterly and monthly growth rates normalized by their standard devia‐tions. Let Zi = {zit}Tt=0, Xi = {Xit}Tt=0, and Si = {Sit}Tt=0 be the i‐th simulation of the unobserved variables, and let 𝜃i be the i‐thsimulation of the parameter vector. The Bayesian procedure follows Carter and Kohn (1994) algorithm, iterating upon Z, X, S,and 𝜃.

1. Given Y, Si, Xi, and 𝜃i, and using the dynamics equation (A.10) and the measurement equation (A.7), simulate Zi+1 usingCarter and Kohn (1994):

a. Initialize z1|0 as a vector with the first element equal to 𝜇0si1 + 𝜇1(1 − si1) + si1xi1 and the remaining elements equal

to zero, and P1|0 = I.

b. Run standard Kalman filter to evaluate {zt|t, Pt|t}Tt=0:

• Account for missing observations in vector yt: define y∗t by eliminating the missing observations, define H∗t by

eliminating the rows that correspond to the missing observations, and define R∗t by eliminating the rows and thecolumns that correspond to the missing observations.

• For t = 1, ..., T, compute the following:

• Forecast:Ωt|t = H∗

t Pt|t−1(H∗t )′ + R∗t ;

𝜈t = yt − H∗t zt|t−1;

Update:

zt|t = zt|t−1 + Pt|t−1(H∗t )′(Ωt|t−1)−1𝜈t;

Pt|t = Pt|t−1 − Pt|t−1(H∗t )′(Ωt|t−1)−1H∗

t Pt|t−1;Prediction:

zt+1|t = Fzt|t + sit+1(𝜇0 + xit+1) + (1− sit+1)𝜇1, 0, ..., 0′;

Pt+1|t = FPt|tF′ + Q.

c. Using the output from the Kalman filter, run the smoothing filter backwards in order to obtain {zt|T, Pt|T}Tt=0:

¹⁸We restrict the individual components for the quarterly series to be white noises in the model, so that the first row of �i is zero for i ≤ Q.

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MAGYAR NEMZETI BANK

• Account for the fact that in the dynamics equation (A.10), shocks in vector 𝜀t do not affect all the state variablescontemporaneously, so that the matrix Q is singular, and Ωt|t−1 is potentially non‐invertible. Define zt by elim‐inating the rows that correspond to zero elements in the vector 𝜀t: according to equation (A.11), this meansdiscarding all elements but the first, the fifth, etc. Similarly, define F by eliminating the corresponding rows, anddefine Q by eliminating the corresponding rows and columns.

• For t = T, Kalman filter has already delivered zt|t = zt|T and Pt|t = Pt|T in step (b). We can use this informationto simulate zi+1

T ∼ 𝒩(zT|T, PT|T), and then eliminate part of its elements, as described above, to get zi+1T .

• Going with t backwards from T− 1 to 1, compute the following:

• Forecast and forecast error:

Ωt+1|t = FPt|t(F)′ + Q;

zt+1|t = Fzt|t + sit+1(𝜇0 + xit+1) + (1− sit+1)𝜇1, 0, ..., 0′;

𝜈t+1 = zi+1t+1 − zt+1|t;

• Use the information from t+ 1 to update the estimates for t:

zt|T = zt|t+1 = zt|t + Pt|t(F)′(Ωt+1|t)−1𝜈t+1;Pt|T = Pt|t+1 = Pt|t − Pt|t(F)′(Ωt+1|t)−1FPt|t;

• Use the obtained information to randomize zi+1t ∼ 𝒩(zt|T, Pt|T);

• Eliminate, as described above, elements of zi+1t to obtain zi+1

t .

2. Given Zi+1, Xi, and 𝜃i, operate upon the equation (A.1) to simulate indicators Si+1 following Carter and Kohn (1994):

a. Going forwards, for t = 1, .., T, compute Pt(st = 0):• The initial unconditional probability of normal state is

P0(s0 = 0) = 1− q2− q− p

.

• For t = 1, ..., T, first compute

Pt−1(st = 0) = Pt−1(st−1 = 0) × p+ 1− Pt−1(st−1 = 0) × (1− q);

and observe that

ft|st = 0 ∼ 𝒩(𝜇0, 𝜎2e ),

ft|st = 1 ∼ 𝒩(𝜇1 + xit, 𝜎2e ).

Then,

Pt(st = 0) =Pt−1(st = 0)𝜙 ft−𝜇0

𝜎e

Pt−1(st = 0)𝜙 ft−𝜇0

𝜎e+ 1− Pt−1(st = 0) 𝜙 ft−𝜇1−xit

𝜎e

.

b. Going backwards for t = T, ..., 1, compute Pt+1(st = 0) = PT(st = 0) and simulate the state indicators using theseprobabilities:

• For the last period, we have PT(sT = 0) from step (a). Simulate si+1T using this probability.

• For t = T− 1, ..., 1, compute the probabilities as follows:

si+1t+1 = 0 ⇒Pt+1(st = 0) = Pt(st = 0) × p

Pt(st = 0) × p+ 1− Pt(st = 0) × (1− q) ;

si+1t+1 = 1 ⇒Pt+1(st = 0) = Pt(st = 0) × (1− p)

Pt(st = 0) × (1− p) + 1− Pt(st = 0) × q.

Then, since Pt+1(st = 0) = PT(st = 0), use this probability to simulate si+1t

3. Given, Si+1, 𝜃i, and Zi+1, simlulate Xi+1 using Carter and Kohn (1994). It is the same routine as in step 1, except that now,the dynamics of the unobserved state are given by equation (A.2), and the “measurement” equation is equation (A.1) forthe common factor, in which all the elements except for xt are fixed. A major simplification is that the common factorvalues are known for all periods t = 1, ..., T, and both the common factor ft and the latent variable xt are one‐dimensional,so there is no need to reduce the dimensionality of the equations to account for missing variables or singularity.

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APPENDIX A MODEL AND ESTIMATION

4. Finally, given Y, Si+1, Zi+1, and Xi+1, compute 𝜃i+1 using standard pior distributions:

a. The prior distribution for 𝜎2v , the variance of the shock affecting the unobserved process xt, is inverse‐gamma: 𝜎2

v ∼ℐ𝒢(a, b). Then, the posterior is also inverse‐gamma, ℐ𝒢(a, b), such that

a = a+ T2 , b = b+

∑(xit − sitxit−1)2

2 .

We sample (𝜎i+1v )2 from this posterior.

b. Similarly, the prior distribution for 𝜎2e , the variance of the shock affecting the common factor ft, is inverse‐gamma:

𝜎2e ∼ ℐ𝒢(a, b). Then, the posterior is also inverse‐gamma, ℐ𝒢(a, b), such that

a = a+ T2, b = 1

b+ ∑ fit − sit(𝜇i1 + xit−1) − (1− sit)𝜇i

02/2

.

We sample (𝜎i+1e )2 from this posterior.

c. The prior distribution for 𝜇0 and 𝜇1 is normal:

𝜇0

𝜇1∼ 𝒩(a,V).

Define Y∗ = {y∗t }, such that y∗t = fit − sit × xit, and X∗ = [1 − Si, Si]. Then, the posterior distribution is also normal,𝒩(a, V), such that

V = V−1 + (X∗)′X∗−1;

a = V V−1a+ (X∗)′Y∗ .

The regime‐specific means 𝜇i+10 and 𝜇i+1

1 are simulated from this posterior.

d. In our specifications, we have worked with only one quarterly variable, GDP growth. We fix the common‐factorloading 𝛾1 for the quarterly GDP growth to be equal to one for identification purposes—this assumption amounts toscaling the common factor. Then, the factor loadings for the monthly variables are interpreted as relative to the unitloading for the GDP growth. In the prior, a factor loading 𝛾j of a monthly indicator j = 1, ...,M is normally distributed:𝛾j ∼ 𝒩(a,V). Then, define yj,t and fj,t as follows:

yj,t = yj,t − 𝜓ij,1yj,t−1 − ... − 𝜓i

j,Pyj,t−P;fij,t = fit − 𝜓i

j,1fit−1 − ... − 𝜓ij,Pfit−P.

We can use these definitions together with equations (A.3) and (A.4) to derive the following expression:

yj,t = 𝛾j fij,t + ej,t, ej,t ∼ 𝒩(0, 𝜎2j ).

Using this expression, we can find the posterior for the factor loading to be normal as well: 𝛾j ∼ 𝒩(aj, Vj), such that

Vj = V−1 + (Xj)′Xj−1;

aj = V V−1a+ (Xj)′Yj ,

where Xj and Yj are vectors with elements {fj,t} and {yj,t} defined above. Factor loadings {𝛾i+1j } are simulated from

these posteriors.

e. We assume that the individual component of the only quarterly variable in our model is a white noise, which makesit simpler to compute the posterior, due to the monthly missing observations in a variable at the quarterly frequency:for the individual component of GDP growth, equation (A.4) reduces to

u1,t = e1,t, e1,t ∼ 𝒩(0, 𝜎21 ).

Then, we specify the inverse‐gammaprior𝜎21 ∼ ℐ𝒢(a, b), which conjugateswith the inverse‐gammaposterior ℐ𝒢(a, b),

such thata = a+ T

2, 1

b+ ∑(yi1,t)2/2.

The variance (𝜎i+11 )2 is simulated from this posterior.

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f. Define Yj = (yj,1, ..., yj,T) to be the vector collecting the observations of a monthly variable j. Let Xj be the T× Pmatrixrecording the P lags of the variable yj,t. For the AR coefficients of each monthly variable’s individual component,𝜓j = (𝜓j,1, ..., 𝜓j,P)′, we assume the same normal prior: 𝜓j ∼ 𝒩(a,V). Then, the posterior is also normal,𝒩(aj, Vj),but different for each j, such that

Vj = V−1a+X′j Xj(𝜎i

j )2

−1

;

aj = Vj V−1a+X′j Yj(𝜎i

j )2

We simulate 𝜓i+1j from this posterior. Finally, to simulate 𝜎i+1

j , we assume inverse‐gamma prior: for each j, 𝜎2j ∼

ℐ𝒢(a, b). LetΨi+1j be the following P × Pmatrix:

Ψi+1j =

⎡⎢⎢⎢⎢⎢⎢⎣

𝜓i+1j,1 ⋯ 𝜓i+1

j,P−1 𝜓i+1j,P

1 ⋯ 0 0

⋮ ⋱ ⋮ ⋮

0 ⋯ 1 0

⎤⎥⎥⎥⎥⎥⎥⎦

Then, we can express the posterior of 𝜎2j as inverse‐gamma distribution as well, ℐ𝒢(a, b), such that

a = a+ T2; b = b+

(Yj − XjΨi+1j )′(Yj − XjΨi+1

j )2

−1

.

We simulate AR coefficients {𝜓i+1j } from these posteriors.

g. For Si+1, let ni+111 count the number of times that the indicator si+1

t has remained at one: ni+111 = ∑T

t=2 1(si+1t = si+1

t−1 =1). Similarly, let ni+1

00 count the number of times it has remained at zero, and ni+110 and ni+1

01 count the number of timesit has switched the value. Then, assuming the same beta prior distribution 𝛽(a, b) for both p and q, the posteriordistribution is of the same shape, 𝛽(a, b). In case of p, the probability of remaining in a normal episode (whenst = 0), is updated with a = a+ n00 and b = b+ n01. For q, the updates are a = a+ n11 and b = b+ n10.

Table 2Moments of the prior distributions of the estimated model parameters

Parameter Meaning Distribution 1st moment 2nd moment

Common factor

p Probability of staying in a normal episode 𝛽(a, b) 90 10

q Probability of staying in an abnormal episode 𝛽(a, b) 90 10

𝜇0 Mean growth rate during normal episode∗ 𝒩(a, b) 0.3 0.04

𝜇1 Mean growth rate during recession∗ 𝒩(a, b) ‐0.3 0.04

𝜎2v Variance of shock to recession‐specific mean ℐ𝒢(a, b) T/10 b = (T/10− 1)/10𝜎2f Variance of shock to common factor ℐ𝒢(a, b) T/10 T/10− 1

Individual components

𝛾i Factor loading for ind. variable i† 𝒩(a, b) 0 0.1

𝜓i,j AR coefficient for ind. variable i and lag j 𝒩(a, b) 0 1

𝜎2i Variance of shock to ind. variable i ℐ𝒢(a, b) T/10 (T/10− 1)/10

The table reports the parameters used to specify each country’s model, along with prior distributions for them, which are either normal, beta, orinverse‐gamma. ∗ For some countries, the mean growth rates for normal and abnormal times are different in the prior, to reflect that they normallyexhibit much higher rates of GDP growth or similar facts. † For GDP growth, the factor loading is set at 1, and not estimated, for identificationpurposes.

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Appendix B Supplementary Figures

Figure 10Euro Area

(a) Probability of low economic activity regime: Constant mean model

2000 2005 2010 2015 2020-3

-2

-1

0

1

2

3

4

0

0.1

0.2

0.3

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1

(b) Probability of low economic activity regime: Time‐varying mean model

2000 2005 2010 2015 2020-3

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1

Note: The blue solid line (right axis) plots the monthly probability of low real activity regime, Pr(st = 1), for the corresponding model, and the greybars (left axis) denote the real GDP quarterly growth.

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MAGYAR NEMZETI BANK

Figure 11Australia

(a) Probability of low economic activity regime: Constant mean model

2000 2005 2010 2015 2020-0.5

0

0.5

1

1.5

2

2.5

3

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

(b) Probability of low economic activity regime: Time‐varying mean model

2000 2005 2010 2015 2020-0.5

0

0.5

1

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2

2.5

3

0

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1

Note: The blue solid line (right axis) plots the monthly probability of low real activity regime, Pr(st = 1), for the corresponding model, and the greybars (left axis) denote the real GDP quarterly growth.

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APPENDIX B SUPPLEMENTARY FIGURES

Figure 12Canada

(a) Probability of low economic activity regime: Constant mean model

1985 1990 1995 2000 2005 2010 2015 2020-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0

0.1

0.2

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1

(b) Probability of low economic activity regime: Time‐varying mean model

1985 1990 1995 2000 2005 2010 2015 2020-2.5

-2

-1.5

-1

-0.5

0

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1

1.5

2

2.5

0

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0.7

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0.9

1

Note: The blue solid line (right axis) plots the monthly probability of low real activity regime, Pr(st = 1), for the corresponding model, and the greybars (left axis) denote the real GDP quarterly growth.

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Figure 13Japan

(a) Probability of low economic activity regime: Constant mean model

1995 2000 2005 2010 2015 2020-5

-4

-3

-2

-1

0

1

2

3

0

0.1

0.2

0.3

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1

(b) Probability of low economic activity regime: Time‐varying mean model

1995 2000 2005 2010 2015 2020-5

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0

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0

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1

Note: The blue solid line (right axis) plots the monthly probability of low real activity regime, Pr(st = 1), for the corresponding model, and the greybars (left axis) denote the real GDP quarterly growth.

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APPENDIX B SUPPLEMENTARY FIGURES

Figure 14United Kingdom

(a) Probability of low economic activity regime: Constant mean model

1985 1990 1995 2000 2005 2010 2015 2020-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0

0.1

0.2

0.3

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0.7

0.8

(b) Probability of low economic activity regime: Time‐varying mean model

1985 1990 1995 2000 2005 2010 2015 2020-2.5

-2

-1.5

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-0.5

0

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1

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2

2.5

0

0.1

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0.9

1

Note: The blue solid line (right axis) plots the monthly probability of low real activity regime, Pr(st = 1), for the corresponding model, and the greybars (left axis) denote the real GDP quarterly growth.

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Figure 15Brazil

(a) Probability of low economic activity regime: Constant mean model

2000 2005 2010 2015 2020-5

-4

-3

-2

-1

0

1

2

3

4

0

0.1

0.2

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1

(b) Probability of low economic activity regime: Time‐varying mean model

2000 2005 2010 2015 2020-5

-4

-3

-2

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0

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1

Note: The blue solid line (right axis) plots the monthly probability of low real activity regime, Pr(st = 1), for the corresponding model, and the greybars (left axis) denote the real GDP quarterly growth.

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APPENDIX B SUPPLEMENTARY FIGURES

Figure 16Mexico

(a) Probability of low economic activity regime: Constant mean model

1985 1990 1995 2000 2005 2010 2015 2020-6

-5

-4

-3

-2

-1

0

1

2

3

4

0

0.1

0.2

0.3

0.4

0.5

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0.7

0.8

(b) Probability of low economic activity regime: Time‐varying mean model

1985 1990 1995 2000 2005 2010 2015 2020-6

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1

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0

0.1

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Note: The blue solid line (right axis) plots the monthly probability of low real activity regime, Pr(st = 1), for the corresponding model, and the greybars (left axis) denote the real GDP quarterly growth.

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Figure 17Russia

(a) Probability of low economic activity regime: Constant mean model

2004 2006 2008 2010 2012 2014 2016 2018 2020-4

-3

-2

-1

0

1

2

3

4

0

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(b) Probability of low economic activity regime: Time‐varying mean model

2004 2006 2008 2010 2012 2014 2016 2018 2020-4

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2

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0

0.1

0.2

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0.5

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1

Note: The blue solid line (right axis) plots the monthly probability of low real activity regime, Pr(st = 1), for the corresponding model, and the greybars (left axis) denote the real GDP quarterly growth.

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APPENDIX B SUPPLEMENTARY FIGURES

Figure 18India

(a) Probability of low economic activity regime: Constant mean model

2000 2005 2010 2015 2020-2

-1

0

1

2

3

4

5

6

0

0.1

0.2

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(b) Probability of low economic activity regime: Time‐varying mean model

2000 2005 2010 2015 2020-2

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Note: The blue solid line (right axis) plots the monthly probability of low real activity regime, Pr(st = 1), for the corresponding model, and the greybars (left axis) denote the real GDP quarterly growth.

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Figure 19China

(a) Probability of low economic activity regime: Constant mean model

1995 2000 2005 2010 2015 20200

0.5

1

1.5

2

2.5

3

3.5

4

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0.7

(b) Probability of low economic activity regime: Time‐varying mean model

1995 2000 2005 2010 2015 20200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Note: The blue solid line (right axis) plots the monthly probability of low real activity regime, Pr(st = 1), for the corresponding model, and the greybars (left axis) denote the real GDP quarterly growth.

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APPENDIX B SUPPLEMENTARY FIGURES

Figure 20South Africa

(a) Probability of low economic activity regime: Constant mean model

1985 1990 1995 2000 2005 2010 2015 2020-3

-2

-1

0

1

2

3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(b) Probability of low economic activity regime: Time‐varying mean model

1985 1990 1995 2000 2005 2010 2015 2020-3

-2

-1

0

1

2

3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Note: The blue solid line (right axis) plots the monthly probability of low real activity regime, Pr(st = 1), for the corresponding model, and the greybars (left axis) denote the real GDP quarterly growth.

MNB WORKING PAPERS 4 • 2020 45

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MAGYAR NEMZETI BANK

Figure 21Receiver Operating Characteristic (ROC) Curve

(a) US: Full‐sample

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

False positive rate

0

0.2

0.4

0.6

0.8

1

Tru

e po

sitiv

e ra

te

Time-varying mean modelConstant mean model

(b) US: Real‐time

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

False positive rate

0

0.2

0.4

0.6

0.8

1

Tru

e po

sitiv

e ra

te

Time-varying mean modelConstant mean model

(a) Euro Area: Full‐sample

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

False positive rate

0

0.2

0.4

0.6

0.8

1

Tru

e po

sitiv

e ra

te

Time-varying mean modelConstant mean model

Note: Chart A shows the ROC curve associated to the in‐sample inferences of US recessions obtainedwith the time‐varying and constantmeanmodels,with areas under the curve of 0.985 and 0.983, respectively. Chart B shows the ROC curve associated to the real‐time inferences of US recessionsobtained with the time‐varying and constant meanmodels, with areas under the curve of 0.887 and 0.817, respectively. Chart C shows the ROC curveassociated to the in‐sample inferences of Euro Area recessions obtained with the time‐varying and constant meanmodels, with areas under the curveof 0.986 and 0.945, respectively. The larger is the area under the curve associated to a given model the more precise it is in identifying recessions. USrecessions correspond to the ones dated by the NBER, and Euro Area recessions correspond to the ones dated by the CEPR. The dashed line in eachchart represents 45∘. For more information about the ROC curve, see Berge and Jordà (2011).

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APPENDIX B SUPPLEMENTARY FIGURES

Figure 22Time‐varying component of mean growth during recessions

U.S. Euro Area

1950 1960 1970 1980 1990 2000 2010 2020

-4

-3

-2

-1

0

1

2×10-3

2000 2005 2010 2015 2020

-5

-4

-3

-2

-1

0

1×10-3

United Kingdom Canada

1985 1990 1995 2000 2005 2010 2015 2020

-15

-10

-5

0

5

×10-4

1985 1990 1995 2000 2005 2010 2015 2020

-6

-5

-4

-3

-2

-1

0

×10-3

Japan Australia

1995 2000 2005 2010 2015 2020

-5

-4

-3

-2

-1

0

×10-3

2000 2005 2010 2015 2020-8

-6

-4

-2

0

2

4

×10-4

Brazil Mexico

2000 2005 2010 2015 2020

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

×10-4

1985 1990 1995 2000 2005 2010 2015 2020

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

×10-3

Russia India

2004 2006 2008 2010 2012 2014 2016 2018 2020

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

×10-3

2000 2005 2010 2015 2020

-8

-6

-4

-2

0

2

4

6

×10-4

China South Africa

1995 2000 2005 2010 2015 2020

-15

-10

-5

0

5

×10-4

1985 1990 1995 2000 2005 2010 2015 2020-25

-20

-15

-10

-5

0

5

×10-4

Note: The figure plots, for each country, the mean (solid blue line) and median (dashed red line) of the simulated distribution of the time‐varyingrecession mean component, u𝜏1 .

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Page 51: REAL‐TIMEWEAKNESS OFTHEGLOBALECONOMY · Contents Abstract 4 1.Introduction 5 2.InferringHeterogeneousRecessions 7 3.AssessingWeaknessAcrossCountries 10 3.1. Data 10 3.2. TheCaseoftheUnitedStates

APPENDIX B SUPPLEMENTARY FIGURES

MNBWorking Papers 2020/4

Real‐Time Weakness of the Global Economy

Budapest, July 2020

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50 MNB WORKING PAPERS 4 • 2020